New comparison theorems for testing the oscillation of solutions of fourth-order differential equations with a variable argument

Osama Moaaz; Asma Al-Jaser; Amira Essam; Osama Moaaz; Asma Al-Jaser; Amira Essam

doi:10.3934/era.2025182

Electronic Research Archive

2025, Volume 33, Issue 7: 4075-4090. doi: 10.3934/era.2025182

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Research article

New comparison theorems for testing the oscillation of solutions of fourth-order differential equations with a variable argument

1.
Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi Arabia
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said, Egypt

Received: 08 January 2025 Revised: 11 June 2025 Accepted: 12 June 2025 Published: 01 July 2025

In this study, we establish new oscillation criteria for the fourth-order delay differential equation. Our main objective is to build upon recent advancements in the study of the oscillatory behavior of second-order equations and extend these findings to higher-order equations. Although fourth-order equations have numerous applications, their study presents significant analytical challenges due to the complex nature of their solutions, which we will discuss in this study. We use the comparison technique with first-order equations in several approaches. Our results show an improvement in the oscillation test due to the development of some monotonic and asymptotic properties of positive solutions. We present a comparison of the new criteria to test their effectiveness, as well as a comparison with previous studies to illustrate the novelty.
- differential equations,
- oscillatory behavior,
- fourth-order equations,
- comparison theorem
Citation: Osama Moaaz, Asma Al-Jaser, Amira Essam. New comparison theorems for testing the oscillation of solutions of fourth-order differential equations with a variable argument[J]. Electronic Research Archive, 2025, 33(7): 4075-4090. doi: 10.3934/era.2025182

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Abstract

In this study, we establish new oscillation criteria for the fourth-order delay differential equation. Our main objective is to build upon recent advancements in the study of the oscillatory behavior of second-order equations and extend these findings to higher-order equations. Although fourth-order equations have numerous applications, their study presents significant analytical challenges due to the complex nature of their solutions, which we will discuss in this study. We use the comparison technique with first-order equations in several approaches. Our results show an improvement in the oscillation test due to the development of some monotonic and asymptotic properties of positive solutions. We present a comparison of the new criteria to test their effectiveness, as well as a comparison with previous studies to illustrate the novelty.

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